The best way is by car. Private transport is going to be quite
expensive.I know that you can take a train to Naples, switch
to another train to Sorrento, and then take a bus to Amalfi,
but it is a long trip. I know that the going rate for a driver
to make a pickup at the airport in Naples is over $100, so I can
not even begin to imagine the price from Rome. Perhaps
a train from Rome to Naples, and then a taxi or private service
to Amalfi.